Sources and sinks of carbon dioxide (CO2) associated with the land surface are often estimated by inverse models. Such models invert atmospheric measurements of CO2 mixing ratio in order to estimate fluxes of CO2 at the land-atmosphere surface. The accuracy of such inversions is limited by the accuracy of models of atmospheric transport. Vertical transport is surprisingly subtle, given that CO2 is a passive tracer. In particular, the vertical transport and the surface flux can conspire to produce a non-uniform diurnal-average profile of CO2, with a surplus near the ground and a deficit aloft. This is sometimes called the ``rectifier effect."

We construct an idealized one-dimensional model of the rectifier effect and solve it analytically. The model equations and solutions depend on a single ``rectifier" parameter. This parameter represents the degree of day-night difference in the eddy diffusivity of the atmosphere.

Using this model, it is relatively straightforward to perform 1D inverse calculations that account for the diurnal rectifier effect. However, the inverse equations reveal that this particular inversion problem has some potential pitfalls. Specifically, a surface measurement alone is insufficient to infer the depth of vertical transport. Rather, measurements at two or more altitudes are required, from a tower for instance. This is because surface CO2 time series are fundamentally ambiguous in this 1D setting: the same time series may arise from strong surface flux and transport, or weak flux and transport. Furthermore, when CO2 is measured at two and only two altitudes, there may remain further ambiguities. If the measurement altitude aloft is too low, and the CO2 mixes deep into the atmosphere, the inverse estimate may be imprecise. If the altitude of measurement is approximately the same as the depth over which CO2 mixes, then the inverse estimate may be non-unique.